相关论文: Non Existence and Strong Ill-Posedness in $H^2$ fo…
In this paper, we prove the asymptotic stability of the incompressible porous media (IPM) equation near a stable stratified density, for initial perturbations in the Sobolev space $H^k$ with any $2<k \in\mathbb{R}$. While it is known that…
We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability…
We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in $\dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and for any…
In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\infty}$ when the initial data are small perturbations of stable profile $g(x_2).$ Consequently, instability can be inferred. Notably,…
In this paper, we consider the 2-D dissipative incompressible porous media (IPM) equation in both supercritical and subcritical cases. The dissipative IPM equation admits a class of special solutions of the form $\rho(x_1,x_2,t)=f(x_2,t)$,…
We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for…
We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on $\bbR^2$ with respect to the buoyancy frequency $N$. Our target density of stratification is the sum…
We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to…
In this paper, we consider a confined physical scenario to prove global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media (IPM) equation. The result is proved using the…
In this paper, we revisit asymptotic stability for the two-dimensional incompressible porous media equation and the Stokes transport system in a periodic channel. It is well-known that a stratified density, which strictly decreases in the…
We consider a special class of infinite energy solutions to the inviscid incompressible porous medium equations (IPM), introduced in Castro-C\'ordoba-Gancedo-Orive [9]. The (IPM) equations then reduce to a one-dimensional nonlocal nonlinear…
We prove probabilistic well-posedness for a 2D viscous nonlinear wave equation modeling fluid-structure interaction between a 3D incompressible, viscous Stokes flow and nonlinear elastodynamics of a 2D stretched membrane. The focus is on…
Recently, Kiselev and Sarsam proposed the following nonlocal transport equation as a one-dimensional analogue of the 2D incompressible porous media (IPM) equation \begin{eqnarray*} \partial_t\rho+u\partial_x\rho= 0,~u=gH_a\rho,…
It was shown recently by Cordoba, Faraco and Gancedo that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework, uses ideas from the theory of laminates, in…
In this paper, we prove a sharp ill-posedness result for the incompressible non-resistive MHD equations. In any dimension $d\ge 2$, we show the ill-posedness of the non-resistive MHD equations in $H^{\frac{d}{2}-1}(\mathbb{R}^d)\times…
An useful approximation for the displacement of two immiscible fluids in a porous medium is the Hele-Shaw model. We consider several liquids with different constant viscosities, inserted between the displacing fluids. The linear stability…
One proves existence and uniqueness of strong solutions to stochastic porous media equations under minimal monotonicity conditions on the nonlinearity. In particular, we do not assume continuity of the drift or any growth condition at…
We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of the porous medium is described by a step function, with a jump discontinuity across…
We establish the existence of smooth, finite-energy solutions to the 2D incompressible porous media equation (IPM), with a compactly supported uniformly smooth source, which develop singularities in finite time.
We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2d Euler equations of the form: \[\partial_t \omega + u\cdot\nabla\omega = R(\omega),\] where $R$ is any non-trivial second order Riesz transform. Namely, we…